Discrete and continuous time cointegration

We give a definition of integration and cointegration in continuous time and study characterizations of those definitions, in particular for CARMA (Continuous time ARMA) processes. We check also that such singularities in CARMA processes may imply unit roots problems in the discretized corresponding process. Then, error correction representations in continuous time are exhibited and discussed, and a general theorem of representation giving a precise description of the singularities is proved. Lastly, we look at other singularities emphasizing the importance of estimating the continuous time model if it is the true one: a one-dimensional noise can generate a two-dimensional regular AR process.